The first recorded observations of magnetic fields set up by currents were those of Oersted, who discovered that a pivoted compass needle, beneath a wire in which there was a current, set itself with its long axis perpendicular to the wire. Later experiments by Biot and Savart, and by Ampere, led to a relation by means of which we can compute the flux density at any point of space around a circuit in which there is a current.
2.3.1 Biot and Savart law Definition
The magnetic field surrounds the current carrying conductor. For a long straight conductor carrying a unidirectional current, the lines of magnetic flux are closed circular paths concentric with the axis of the conductor. Biot and Savart deduced, from the experimental study of the field around a long straight conductor, that the magnetic flux density B associated with the infinitely long current carrying conductor at a point P which is at a radial distance r, as illustrated in FIG. 2.14, is
r B I
π µ
2= 0 , (2.19)
Where B = magnetic flux density µ0 = permeability of free space I = current
r = radial distance
Unlike the electric field around a charged wire, which is radial, the lines of magnetic induction are circles concentric with the wire and lying in planes perpendicular to it. The direction of this concentric closed loop of magnetic lines is given by right hand rule.
FIG. 2.14. Magnetic field around a long straight current conductor carrying.
Practical rules
Consider a conductor through which current I is flowing vertically upward, as shown in FIG. 2.14. Suppose we want to find the magnetic flux density due to this current carrying conductor at a point P whose perpendicular distance from the conductor is r. It was found experimentally by Biot and Savart that at any point:
(a) the magnetic flux density is directly proportional to the current. i.e. B ∝ I.
(b) the magnetic flux density is directly proportional to the effective length of the conductor, i.e. B ∝ l.
(c) iii) the magnetic flux density is inversely proportional to square of the distance r of point P from the conductor, i.e. B ∝ 1/r2 .
Combining the above three statements the magnetic flux density comes out to be as given by the equation (2.19).
Right hand rule
The lines of magnetic induction are circles concentric with the wire and lying in planes perpendicular to it. The direction of this concentric closed loop of magnetic lines is given by right hand rule, which states, ‘If the conductor is grasped in the right hand with the thumb pointing in the direction of the current, the curled fingers of the hand will point in the direction of the magnetic field’ as shown in FIG. 2.15.
FIG. 2.15. Fleming’s right hand rule.
2.3.2 Ampere’s law Definition
The Ampere’s law states that “the magnetic flux density over a closed surface is directly proportional to the current enclosed by the surface.”
The equation 2.19 can be written as 2πrB = µoI. This shows that the product of 2π r and B equals µoI. But 2πr is the length of the path around the conductor and on it the value of B is the same at every point. Ampere generalized this result into a law. Mathematically, it can be written as
I l
B• =µ0 (2.20)
where
l = length
B = magnetic flux density µ0 = permeability of free space I = current
This law is applicable to closed paths other than circular ones.
Thus Ampere’s law can also be defined as “The dot product of B and l around any closed path equals µo I, where I is the total steady current threaded by the path.”
The path is divided into several small length elements as shown in FIG. 2.16. Consider a length element dl of the closed path. According to Ampere’s law it can be written as
I Bdl=µ0
∫
(2.21)where
dl = length
B = magnetic flux density µ0 = permeability of free space I = current
FIG. 2.16. Verification of Ampere’s law for long straight conductor geometry.
Applications
(a) Field due to current in a toroid
A toroid is a solenoid that has been bent into a circle. Consider a toroid of radius r, having N turns and current I flowing through it. When current passes through each turn, circular magnetic lines of force pass through the turns of the toroid.
Applying Ampere’s law along the axis of the toroid, we can write
∫
B.dl = µo × (current enclosed) = µo NIAs the angle between B and dl is zero and
∫
dl = 2πr, therefore the magnetic field due to current in a toroid according to Ampere’s law is given byr B NI
π µ
2
= 0 (2.22)
where
B = magnetic flux density µ0 = permeability of free space N = number of turns
I = current
r = radial distance
The magnetic filed is not uniform over a cross-section of core, because the path length l is larger at the outside of the section than at the inner side. However, if the radial thickness of the core is small compared to the toroid radius r, the field varies only slightly across a section.
In this case, considering that 2πr is the circumference length of the toroid and that n = N/ 2πr is the number of turns per unit length, the field may be written as
nI
B=µ0 (2.23)
where
B = magnetic flux density µ0 = permeability of free space n = number of turns/unit length I = current
The equations derived above for the field in a closely wound solenoid or toroid are strictly correct only for a winding in vacuum. For most practical purposes, however, they can be used for a winding in air, or on core of non-ferromagnetic material.
(b) Field due to current in a coil
A coil is constructed by winding wire in a helix around a cylindrical surface. The turns of the winding are ordinarily closely spaced and may consist of one or more layers. When it is connected to a battery an electric current flows through each turn and produce magnetic field.
This magnetic field is fairly uniform and stronger inside the turns but it is weaker and negligible outside the coil.
To find the magnetic field due the coil we apply the Ampere’s law. The field outside the coil is zero and the field inside the coil is uniform and stronger and is along the axis of the coil.
Therefore by Ampere’s law I
dl B• =µ0
∫
The value of B will be same as in the case of a toroid, i.e.
nI B=µ0
The direction of B is along the axis of the coil.